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Math Help - Average succesive distances between primes below n, graphed

  1. #1
    Member mfetch22's Avatar
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    Average succesive distances between primes below n, graphed

    I wrote a computer program function that takes an integer as input and outputs the average distance between all the succesive primes below this integer. I graphed the data and ran a logarithmic regression on it and came up with the green curve seen in the attached image. Would this regularity be something expected of a graph like this? Its just an interesting property, to me.
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  2. #2
    MHF Contributor chiph588@'s Avatar
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    Quote Originally Posted by mfetch22 View Post
    I wrote a computer program function that takes an integer as input and outputs the average distance between all the succesive primes below this integer. I graphed the data and ran a logarithmic regression on it and came up with the green curve seen in the attached image. Would this regularity be something expected of a graph like this? Its just an interesting property, to me.
    The prime number theorem tells us  \pi(x)\sim\frac{x}{\log x} .

    This can be manipulated to show  p_n\sim n\log n , where  p_n is the  n^{th} prime.

    So loosely speaking,  S(n)=\frac{1}{\pi(n)}\sum_{i=2}^{\pi(n)} (p_i-p_{i-1})=\frac{p_{\pi(n)}-2}{\pi(n)}  \sim \frac{\pi(n)\log \pi(n)}{\pi(n)}=\log \pi(n) \sim \log n-\log\log n \sim \log n since the sum is telescoping.
    Last edited by chiph588@; June 9th 2010 at 02:19 PM.
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